Refactorization of Cauchy’s Method: A Second-Order Partitioned Method for Fluid–Thick Structure Interaction Problems

Bukač, Martina; Seboldt, Anyastassia; Trenchea, Catalin

doi:10.1007/s00021-021-00593-z

Cited by 13 publications

(11 citation statements)

References 40 publications

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“…The (DLN) methods are indexed by the free parameter δ ∈ [0, 1]. When δ = 1, the (DLN) method becomes the (implicit) midpoint rule [7,8]…”

Section: The Dln Methods and Its Refactorizationmentioning

confidence: 99%

“…The refactorization of an algorithm to reduce its cognitive complexity has been used in [7] to rearrange a family of one-leg one-step methods into a backward Euler code followed by post-processing, and further applied for partitioning multi-physics problems [17,18,8,19]. In [20], the authors describe the implementation of the (DLN) formulas in a Nordsieck formulation [21,22] essentially identical to that of the backward differentiation formulas, facilitating to adapt Nordsieck formulation codes like DIFSUB [23,24] to the (DLN) formulas.…”

Section: Related Workmentioning

confidence: 99%

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Refactorization of a variable step, unconditionally stable method of Dahlquist, Liniger and Nevanlinna

Layton¹,

Pei²,

Trenchea³

2021

Preprint

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The one-leg, two-step time-stepping scheme proposed by Dahlquist, Liniger and Nevanlinna has clear advantages in complex, stiff numerical simulations: unconditional G-stability for variable time-steps and second-order accuracy. Yet it has been underutilized due, partially, to its complexity of direct implementation. We prove herein that this method is equivalent to the backward Euler method with pre-and post arithmetic steps added. This refactorization eases implementation in complex, possibly legacy codes. The realization we develop reduces complexity, including cognitive complexity and increases accuracy over both first order methods and constant time steps second order methods.

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“…The (DLN) methods are indexed by the free parameter δ ∈ [0, 1]. When δ = 1, the (DLN) method becomes the (implicit) midpoint rule [7,8]…”

Section: The Dln Methods and Its Refactorizationmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Refactorization of a variable step, unconditionally stable method of Dahlquist, Liniger and Nevanlinna

Layton¹,

Pei²,

Trenchea³

2021

Preprint

Self Cite

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“…The one-leg 2-step (DLN) methods (8) satisfy unconditionally the following long-time energy bounds: for any integer M > 1, (8). Using the skew-symmetry property of b and the Cauchy-Schwarz inequality we obtain…”

Section: Theoremmentioning

confidence: 99%

“…Theorem 2 Let (u(t), p(t)) be a sufficiently smooth, strong solution of the (NSE). Under the time step condition (11), there exists a constant C > 0 such that the solution to the DLN algorithm (8) satisfies the following error estimates…”

Section: Lemma 2 Let U(t) Be Any Continuous Function Onmentioning

confidence: 99%

“…The accurate numerical simulation of flows of an incompressible, viscous fluid, with the accompanying complexities occurring in practical settings, is a problem where speed, memory and accuracy never seem sufficient. For time discretization (considered herein), many longer time simulations use constant step low order methods, and (with few exceptions noted in Section 1.1) the remainder use the constant time step implicit midpoint or the trapezoidal schemes, for example, [2,3,8,31,39], (often combined with fractional steps, or with ad hoc fixes to correct for oscillations due to lack of L-stability [6,38,44]), or the backward differentiation formula 2 (BDF2) method [1,18,22,33,34,46]. Time accuracy requires time step adaptivity within the computational, space and cognitive complexity limitations of computational fluid dynamics (CFD).…”

Section: Introductionmentioning

confidence: 99%

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Analysis of the variable step method of Dahlquist, Liniger and Nevanlinna for fluid flow

Layton

Pei

Qin

et al. 2021

Numerical Methods Partial

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The one-leg, two-step time discretization proposed by Dahlquist, Liniger and Nevanlinna is second order and variable step G-stable. G-stability for systems of ordinary differential equations (ODEs) corrresponds to unconditional, long time energy stability when applied to the Navier-Stokes equations (NSEs). In this report, we analyze the method of Dahlquist, Liniger and Nevanlinna as a variable step, time discretization of the Navier-Stokes equations. We prove that the kinetic energy is bounded for variable time-steps, show that the method is second-order accurate, characterize its numerical dissipation and prove error estimates. The theoretical results are illustrated by several numerical tests.

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An hp Error Analysis of HDG for Linear Fluid–Structure Interaction

Meddahi

2025

J Sci Comput

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A variational formulation based on velocity and stress is developed for linear fluid–structure interaction problems. The well-posedness and energy stability of this formulation are established. A hybridizable discontinuous Galerkin method is employed to discretize the problem. An hp-convergence analysis is performed for the resulting semi-discrete scheme. The Crank–Nicolson method is used for temporal discretization, and the convergence properties of the fully discrete scheme are examined. Numerical experiments are presented to validate the theoretical results and demonstrate the accuracy of the proposed method.

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Refactorization of Cauchy’s Method: A Second-Order Partitioned Method for Fluid–Thick Structure Interaction Problems

Cited by 13 publications

References 40 publications

Refactorization of a variable step, unconditionally stable method of Dahlquist, Liniger and Nevanlinna

Refactorization of a variable step, unconditionally stable method of Dahlquist, Liniger and Nevanlinna

Analysis of the variable step method of Dahlquist, Liniger and Nevanlinna for fluid flow

An hp Error Analysis of HDG for Linear Fluid–Structure Interaction

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